Optimal. Leaf size=64 \[ \frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{b} \sqrt{a^2-b^2 x^4}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0691381, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{b} \sqrt{a^2-b^2 x^4}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a - b*x^2]/Sqrt[a^2 - b^2*x^4],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 12.7305, size = 56, normalized size = 0.88 \[ \frac{\sqrt{a^{2} - b^{2} x^{4}} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{\sqrt{b} \sqrt{a - b x^{2}} \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-b*x**2+a)**(1/2)/(-b**2*x**4+a**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0280174, size = 67, normalized size = 1.05 \[ \frac{\log \left (\sqrt{b} \sqrt{a-b x^2} \sqrt{a^2-b^2 x^4}+a b x-b^2 x^3\right )-\log \left (b x^2-a\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a - b*x^2]/Sqrt[a^2 - b^2*x^4],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.015, size = 54, normalized size = 0.8 \[{1\sqrt{-{b}^{2}{x}^{4}+{a}^{2}}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{-b{x}^{2}+a}}}{\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-b*x^2+a)^(1/2)/(-b^2*x^4+a^2)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-b*x^2 + a)/sqrt(-b^2*x^4 + a^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.276606, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{2 \, \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} b x -{\left (2 \, b^{2} x^{4} - a b x^{2} - a^{2}\right )} \sqrt{b}}{b x^{2} - a}\right )}{2 \, \sqrt{b}}, -\frac{\arctan \left (\frac{\sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} \sqrt{-b}}{b^{2} x^{3} - a b x}\right )}{\sqrt{-b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-b*x^2 + a)/sqrt(-b^2*x^4 + a^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a - b x^{2}}}{\sqrt{- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x**2+a)**(1/2)/(-b**2*x**4+a**2)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-b x^{2} + a}}{\sqrt{-b^{2} x^{4} + a^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-b*x^2 + a)/sqrt(-b^2*x^4 + a^2),x, algorithm="giac")
[Out]